I enjoy the @nd Avenue Deli but I'm a Katz's man all the way. Love their Reuben. There are definitely more than three in Manhattan though Katz's, 2nd Ave and carnegie are the most well known and get the most press.
Now that Stage and Sarge's are gone, I don't know of any others.....I understand that decades ago there were over 300 or 400 just on the Lower East Side.
I know there are only about 4 or 5 left on Long Island. I don't think there are any left in Southwestern CT. and probably about 3 or 4 each in Northern NJ and Rockland County ( I don't think Westchester has any left)
I forgot about the Boroughs, maybe a handful (if that many)